The Eigenvalue Estimations Of Hadamard Product Of Nonnegative Matrix And M Matrix

The estimation of matrix eigenvalue is one of the important problemof matrix theory.Based on the existing literature,it is gived that the upperbounds of the lagest eigenvllue of nonnegative matrix Hadamard productused the estimation method of matrix eigenvalue on the oval area andlower bounds estimation of minimum eigenvalue of Hadamard product ofa M matrix and the inverse matrix of M matrix used the maximum of theline addition except the main diagonal elements.The text conclude five chapters content is as follows:In chapter1,the bachground and the reseach status of nonnegativematrix and the M matrix are gived,and the Hadamard product ofnonnegative matrix and M matrix research status and applicationbackground are instructed.In chapter2,this paper introduces some definitions used in thisartrcle,and some conclusions and theorem used in this article.The atfirst,it is introduced that the definitions of nonnegative matrix,matrix andHadamard product,spectral radius and reducible matrix;and then it isintroduced that some familiar theorem used in this paper.In chapter3, used the estimation method of matrix eigenvalues on the oval area, the upper bounds of the largest eigenvalue of the two nonnegative matrix Hadamard product was studied. New estimation are given as follows:Any of positive vectors α1, α2,…αn are given, D=diag (α1, α2,…αn), the new estimate of largest eigenvalue was proved.Through numerical example, it was found that the estimation formula of this paper is more accurate than the existing literature.And the new estimation formula of calculation only relyed on the matrix elements and matrix F norm,and it is easy to calculate relatively.In chapter4,on the basis of the existing literature,the two new estimation of the smallest eigenvalue of Hadamard product of M matrix and the inverse matrix of other M matrix is given: q(AΟB)≥min i{aiibii-mi[ρ(B)-bii]}(mi=max j≠1{|aij|}), q(AΟB)≥min i{aiibii-mi[aii-q(A)]}(mi=max j≠1{|bij|}),On the basia of the estimation above,a new estimation of the smallest eigenvalue of two nonsingular M matrix’Hadamard product: q(AΟB-1)≥min i{aiicii-[ρ(B-1)-cii][aii-q(A)]}.Through the example,it was found that although the estimation formula is no more accurate than existing estimate type,multiple estimate type used at the same time can narrow the scope of the smallest eigenvalue for some specific matrix. In chapter5,the conclusion was gived in this article,and the researchdirection was put forward.